XLVII XLVIII] Introduction ccxxix 



/^Z 



a; = 0, w = A / 

 V m 



1, w + 1) should be =1. 



Hence we have an useful criterion for the effectiveness of this formula. It will be 

 found in many cases by no means satisfactory, if x' = x be greater than 1 to 1*5 

 times the standard deviation. 



Thus we are compelled to admit that while formula (xii) is good for areas at 

 the tails and formula (xiii) for areas round the mode, neither is satisfactory for 

 areas having a bounding ordinate between \a (or l*5<r) and 3<r (or 2'5a) from the 

 mode, that is to say from the mathematical standpoint which may demand six to 

 seven figure accuracy. We may also remark that, even with the present Tables, 

 both formulae require very considerable arithmetical labour. 



A method which can give good results when I and m are small, is due to Soper*. 

 By taking our origin at one or other end of the curve, we can be certain that x is 

 ^ . We then represent the binomial (1 x) m by a quintic polynomial passing 

 through the six points equally spaced from x = to : 



2/o 2/i 2/2 2/3 2/4 2/5 I 

 1 ("9) m (-8)" (-7) w (-6) ('oH ' 



The quintic f is 

 3fe = -rio [ 12 2/o - l (274yo - 6<% x + 6007/ 2 - 4007/3 + 150y 4 - 24y 6 ) 



1070y 2 - 7HOy 3 + 305y 4 - 



- 120y 3 + 55y 4 - 10y 6 ) 

 - (^ (2/0-57/1 + 1 Of/a - 107/3 + 5y 4 - 7/5) L 



where in our particular case ^ = i x i = 



* toe. c. pp. 2122. He suggests that when m is large, it may be possible to obtain a good result 

 by fitting the quintic only to the range of integration (p. 28). 



t The six j/'s may of course be calculated for other functions than (1 - x) m and the integral 



x l f(x)dx, ie. the Hh incomplete moment function of y =/(*), thus determined. 



